The properties of limit are the following:
Sum of limitsproduct ruleDifference ruleConstant multiply rulelaw of constant, etc .A limit is a value which obtained when a function approaches the output value for an input value. For a variable x approaches a particular value say "a", i.e., x → a and limit is denoted as [tex] \lim_{x \to a} f(x) \\ [/tex].
The basic properties of limit are :
1) Sum rule : limit of sum of two functions is equal to sum of individual function limit.
[tex]\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a}g(x) \\ [/tex]
2) Product rule : The limit of product of two functions is equal to product of individual function limits.
[tex]\lim_{x \to a} (f(x) \times g(x)) = \lim_{x \to a} f(x) \times lim_{x \to a} g(x) \\ [/tex]
3) Difference rule : The limit of difference of two functions is equal to difference of individual function limits.
[tex]\lim_{x \to a} (f(x)-g(x))=\lim_{x \to a} f(x) - \lim_{x \to a}g(x) \\ [/tex]
4) Law of constant : The limit of a constant is equal to constant itself.
[tex]\lim_{x \to a}c \: = c \\ [/tex]
5) Constant multiply rule : The limit of product of constant with function is equal to constant times limit of function.
[tex]\lim_{x \to a} (k \times f(x)) = k\times lim_{x \to a}f(x)\\ [/tex]
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A triangle with angles of 50°, 60°, and 70° is inscribed in a circle of radius 8 inches. Find the lengths of all three sides of the triangle.
The lengths of all three sides of the triangle are;
43.6 inches
52.3 inches
61.1 inches
What are the lengths of the triangle?The arcs are seen to be two times the angles and as such, the arc made by the 50° angle is 100° and then we see that the one made by the 60 degree angle given as 120°.
Formula for the circumference of a circle is;
C = 2πr
C = 50π inches
C = 157 inches
The arc made by the 50° angle is;
Arc length = 157*100/360
= 43.6 inches
The arc length made by the 60° angle = 157(120/360)
= 52.3 inches
The arc length made by the remaining 70° angle is;
Arc length = 157(140/360)
Arc length = 61.1 inches
Total arc length = 43.6 + 52.3 + 61.1
= 157 inches.
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Solvent and graphing systems of equationsm
According to Maths, in order to solve a system of linear equations by graphing, we must first graph both equations in the same coordinate plane and then locate the intersection of the two lines.
What is a System of Equations?A system of equations in algebra consists of two or more equations and looks for common answers to the equations. "A set of equations satisfied by the same set of variables is called a system of linear equations."
According to the discussion above, a system of equations is a collection of equations that aim to find a common solution for all of the variables. An illustration of a system of equations is the set of equations below.
2x - y = 12
x - 2y = 48
Finding the values of the variables employed in a system of equations entails solving the set of equations. While keeping the equations balanced on both sides, we compute the values of the unknown variables. Finding the value of the variable that makes the condition of all the given equations true is the primary goal when solving an equation system. A given system of equations may have a variety of solutions,
Unique solutionNo solutionInfinitely many solutionsSolving System of EquationsThere are numerous ways to solve any set of equations. We require at least two equations in order to solve a system of two-variable equations. Similar to this, we will need at least 3 equations to solve a system of 3 equations. Let's examine three approaches to solving a set of equations, presuming that they are linear equations in two variables.
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Help fast please! (Find the value of y.)!!!
Answer:
y = 10 and x = 5
Step-by-step explanation:
using the sine ratio in the right triangle and the exact value
sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , then
sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{5\sqrt{3} }{y}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
y × [tex]\sqrt{3}[/tex] = 10[tex]\sqrt{3}[/tex] ( divide both sides by [tex]\sqrt{3}[/tex] )
y = 10
-----------------------------------
using Pythagoras' identity in the right triangle
x² + (5[tex]\sqrt{3}[/tex] )² = y² = 10²
x² + 75 = 100 ( subtract 75 from both sides )
x² = 25 ( take square root of both sides )
x = [tex]\sqrt{25}[/tex] = 5
How do you prove 3 angles of a triangle is 180?.
If the two lines crossed are parallel, the alternate angles are equal. It is proven that the sum of the three angles of a triangle is equal to 180 degrees.
We have to prove that ∠A + ∠B + ∠C is equal to 180 degrees.
The measure of A = ∠1
B = ∠2
C = ∠3
Given, l || BC
We know that the alternate angles are equal in parallel lines.
∠2 = ∠4 -------------- (1)
∠3 = ∠5 -------------- (2)
Adding (1) and (2),
∠2 + ∠3 = ∠4 + ∠5
By adding ∠1 on both sides,
∠1 + ∠2 + ∠3 = ∠4 + ∠5 + ∠1
From the figure,
The sum of the angles at a point on a line is equal to 180 degrees.
∠4 + ∠1 + ∠5 = 180°
So, ∠1 + ∠2 + ∠3 = 180°
Therefore, ∠A + ∠B + ∠C = 180°
Thus, If the two lines crossed are parallel, the alternate angles are equal. It is proven that the sum of the three angles of a triangle is equal to 180 degrees.
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Marcus buys 7 3/4 feet of wrapping paper. He lets his sister use 2 1/3 feet if the wrapping paper to wrap a gift. How much wrapping paper does Marcus have left?
The wrapping paper Marcus have left is 5 5 / 12 feet.
How to find the wrapping paper Marcus have left?Marcus buys 7 3/4 feet of wrapping paper. He lets his sister use 2 1/3 feet of the wrapping paper to wrap a gift.
The amount of wrapping paper Marcus have left can be calculated as follows:
The amount of wrapping paper Marcus have left is the subtraction of the paper used by the sister by the wrapping paper he had initially.
Hence,
wrapping paper Marcus have left = 7 3 / 4 - 2 1 / 3
wrapping paper Marcus have left = 31 / 4 - 7 / 3
wrapping paper Marcus have left = 93 - 28 / 12
wrapping paper Marcus have left = 65 / 12
Therefore,
wrapping paper Marcus have left = 5 5 / 12 feet
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If you are paid $236. 64 for 24 1/2 hour of work what amount hould you be paid for 32 hour of work at thi time ame rate to pay
I would be paid $310.4 for 32 hour of work at the same rate
What are algebraic operations?They are the set of numbers and symbols that are related by the different mathematical operation signs such as addition, subtraction, multiplication, division among others.
To solve this problem we must perform the following algebraic operations with the given information
Information about the problem:
Hours worked = 24 1/2 hPaid = $236.64Hours = 32 hPaid (32h)= ?We transform the mixed fraction into improper fraction and we have:
24 1/2 =
[(24*2) + 1)] /2=
[48+1] /2=
49/2
Calculating the amount of dollars paid for 1 hour worked:
Dollar per hour = Paid / Hours worked
Dollar per hour = $236.64 / (49/2)
Dollar per hour = $236.64 * 2 / 49
Dollar per hour = $473.28/49
Dollar per hour = $9.70
Calculating the amount of dollars paid for 32 hours worked:
Paid (32h) = Dollar per hour * Hours
Paid (32h) = $9.70/h * 32 h
Paid (32h) = $310.4
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DRIVING While on a bike ride, Tyra’s distance from home can be modeled by f(x) = −15∣∣x−40∣∣+8
, where x is the time since Tyra left home in minutes and f(x) her distance from home in miles. Graph the function on a separate piece of paper. Find and interpret the key features of the graph in the context of the situation.
The graph of the function f(x) = −15|x−40|+8 will be a piecewise linear function.
Graph explain: What is it?
A graph is defined as a mathematical structure that connects a collection of points to express a specific function. A pairwise relationship between the objects is established using it. The edges that connect the graph's vertices (also known as nodes) are its vertices (lines).
It will be a V-shape, with the vertex at x = 40 and y = 8.
On the left side of the vertex, the function will be decreasing, representing Tyra moving away from home.
On the right side of the vertex, the function will be increasing, representing Tyra moving towards home.
The vertex at x = 40 and y = 8 represents the point in time where Tyra reaches her furthest distance from home (8 miles) and then begins to head back towards home.
Key features of the graph in the context of the situation:
The vertex at x = 40 and y = 8 represents the point in time where Tyra reaches her furthest distance from home (8 miles) and then begins to head back towards home.
The left side of the vertex represents the time period when Tyra is moving away from home, and the right side represents the time period when Tyra is moving towards home.
The slope of the graph on the left side of the vertex is negative, indicating that Tyra is moving away from home at a decreasing rate, while the slope of the graph on the right side of the vertex is positive, indicating that Tyra is moving towards home at an increasing rate.
The y-intercept is at 8, which means Tyra starts 8 miles away from home.
The graph is symmetric about the y-axis because for every time x on the left side of the vertex, there is a corresponding time on the right side of the vertex.
The graph is not a function because for every x value, there are two y values, one for when Tyra is moving away from home and one for when Tyra is moving towards home.
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Please solve the question correctly
Answer:
5x + 7y + 4
Step-by-step explanation:
5x + (6y+y) ( 3+1)
5x + 7y + 4
HELP ME PLEASE. I dont get this
The value of x for the given angles is 23
What is transversal?A transversal is a line that, at two different locations in the geometry, cuts through two lines in the same plane. The angles that result from a transversal crossing of two lines in pairs can be successive internal angles, matching angles, or alternate angles.
Given lines m and n, where m ║ n
and t intersect at both the lines as shown in the figure,
so line t works s transversal,
and angles,
(8x - 27)° and (7x - 4)°
both the angles are at the bottom of the line and same side so they are corresponding angles,
and corresponding angles for any transversal are equal,
(8x - 27)° = (7x - 4)°
8x - 27 = 7x - 4
8x - 7x = -4 + 27
x = 23
Hence the value of x is 23.
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Zayn has a garden that is 86 metres long and x metres wide. He decides to build a fence around the garden. If the perimeter of the garden is 268 m, what is the width? Remember to write the algebraic expression.
Step-by-step explanation:
the perimeter of a rectangle is
2×length + 2×width
in our case we have
268 = 2×86 + 2x = 172 + 2x
96 = 2x
x = 96/2 = 48 m
2. Finds
Geometry - Clark
Midpoint and Distance Formulas
© 2011 Kuta Software LLC. All rights reserved.
Find the midpoint of the line segment with the
1) (-4,-2), (3, 3)
The midpoint of the line segments with
1) (-4,-2), (3, 3) is ( -1/2 , 1/2).
2) (−1, −6), (−6, 5) is ( -7/2 , -1/2 ).
3) (2, 4), (1, −3) is ( 3/2 , -1/2).
In geometry, the midpoint is defined as the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. To determine the midpoint, just add the two X-values, then divide by 2 and add the two Y-values, then divide by 2. Mathematically, let (xm , ym) be coordinates of mid point of line segment ents with end points ( x₁, y₁) , (x₂, y₂). Then, xₘ =( x₁ + x₂)/2 and yₘ = ( y₁ + y₂)/2
We have , the line segments with coordinates
1) (-4,-2), (3, 3)
Midpoint = ( (-4 + 3)/2 , (-2 + 3/2) )
= ( -1/2 , 1/2)
2) (−1, −6), (−6, 5)
Midpoint = ( ( -6 +(-1))/2 , ( -6 + 5)/2)
= ( -7/2 , -1/2 )
3) (2, 4), (1, −3)
Midpoint = ((2 + 1)/2 , (4+(-3))/2)
= ( 3/2 , -1/2)
Hence, we got all midpoints for line segments.
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Complete question:
Geometry - Clark Midpoint and Distance Formulas
© 2011 Kuta Software LLC. All rights reserved.
Find the midpoint of the line segment with the given endpoints.
1) (-4,-2), (3, 3)
2) (−1, −6), (−6, 5)
3(2, 4), (1, −3)
How should you graph the equation y =- 3 4x 2?.
The final graph is given below in the image.
The equation is [tex]y = -\frac{3}{4}x - 2[/tex]
The graph of a linear equation in two variables is a line (that's why they call it linear).
If you know an equation is linear, you can graph it by finding any two solutions
(x1,y1) and (x2,y2)
plotting these two points, and drawing the line connecting them.
You can find two solutions, corresponding to the x-intercepts and y-intercepts of the graph, by setting first x=0 and then y=0.
To graph the line [tex]y = -\frac{3}{4}x - 2[/tex] we are going to take advantage of the fact that its y-intercept is -2 when x = 0, so our first point is (0,-2). To find our second point, we are going to find the x-intercept; to do it, we are going to set y = 0 and solve for x:
[tex]y = -\frac{3}{4}x - 2\\0 = -\frac{3}{4}x - 2\\2 = -\frac{3}{4}x \\x = -\frac{8}{3}[/tex]
Now, we have our second point (0, -8/3), we just need to join the 2 points with a straight line.
The final graph is given below in the image.
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Without knowing its side lengths, can you determine the ratio of the perimeter of a square to the length of one of its sides? Explain.
A square's side to perimeter ratio is one to four.
How do you calculate a perimeter's ratio?The quantity of two-dimensional space a shape occupies is referred to as its area. Simply dividing the perimeter by the area yields the ratio of a shape's perimeter to area. It is simple to calculate this. The perimeter ratio and scale factor are the same. In reality, the scaling factor for every ratio between any two similar shapes is the same (diagonals, medians, midsegments, altitudes, etc.).The Perimeter is equal to a constant multiplied by the Side length, or the Perimeter is equal to four when divided by the Side length. There is so no doubt that the link between Side length and Perimeter is proportionate.To learn more about ratio, refer to:
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A diagonal of a rectangle splits the rectangle into two 300-60 •-90° triangles. If the diagonal of the rectangle is 21 in, what is the length and width of the rectangle. Find the area
When the diagonal of a rectangle splits the rectangle into two producing angles 30 - 60 - 90 the sides of the rectangle is
Area of a rectangle is solved to get 191 square in
How to find the lengths of the rectangleThe problem is solved using SOH CAH TOA
Sin = opposite / hypotenuse - SOH
Cos = adjacent / hypotenuse - CAH
Tan = opposite / adjacent - TOA
The shape describes a right triangle of
opposite = width
adjacent = length
hypotenuse = diagonal
The dimensions are calculated using SOH and CAH
sin 30 = opposite / hypotenuse
sin 30 = width / 21
width = 21 * sin 30
width = 10.5
For the length
cos 30 = adjacent / hypotenuse
cos 30 = length / 21
length = 21 * sin 30
length = 18.19
Area of a rectangle = length * width
Area of a rectangle = 18.19 * 10.5
Area of a rectangle = 190. 995
Area of a rectangle = 191 square in
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Complete the statement with the correct value. When z = -4. -2z + 1 = A. 6. B. 7. C. 9
The statement with the correct value is C. 9
What are simple substitutions?Simple substitution is a topic which are mathematical expressions with variables whose value/s has been given. Thus it is just required to solve the question by simple substitution of the given value in the expression. Example, given that a = 2, determine the value if a + 6.
This can be solved by substituting 2 for a in the question, so that;
a + 6 = 2 + 6
= 8
Therefore, it can be deduced from the given question that;
-2z + 1, but z = -4
Thus,
-2z + 1 = -2(-4) + 1
= 8 + 1
= 9
To complete the statement, the correct value is 9. Option C.
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What are the 3 defining characteristics of parallel lines?.
The characteristics of parallel lines:
1) Parallel lines lie in the same plane
2) The parallel lines never intersect each other.
3) The slopes of parallel lines are equal.
We know that two lines are parallel to each other if they lie in the same plane and never intersect each other.
Parallel lines are always equidistant from each other.
Also, the parallel lines are coplanar.
Mathematically, the parallel lines are represented by the symbol, ‘||‘.
If two lines l₁ and l₂ are parallel to each other then they are represented as l₁ || l₂.
Also, the slopes of parallel lines are equal.
Let m₁ be the slope of line l₁ and m₂ be the slope of line l₂.
If line l₁ and l₂ are parallel to each other then m₁ = m₂
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What is isosceles angle theorem?.
Isosceles Triangle Theorem: If two triangle sides are similar, then their corresponding angles must also be similar. The sides of a triangle that correspond to two angles that are congruent are also congruent.
According to the isosceles Angle Theorem, the angles in an isosceles triangle that face congruent sides are also congruent.
A triangle with two equal sides is said to be isosceles. Isosceles triangle is basically such triangle which have two sides equal in magnitude. It is to be noted that because two angles should also be equal because of the two equal sides.
Angles opposing two triangle sides are congruent if their two sides are as well.
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The value of a motorcycle is depreciating by 2.8% each year. If the bike was purchased for $12,000, how much will it be worth in 12 years?
Type your answer as a decimal rounded to the nearest cent.
Answer:8534.46
Step-by-step explanation:
The depreciated value of the bike after 12 years is $8534.46
What is depreciated value?Depreciated value is the minimum amount that an asset is worth after depreciation.
Given that, the value of a motorcycle is depreciating by 2.8% each year, the bike was purchased for $12,000,
We know that, the depreciation value is given by,
A = P(1-r)ⁿ
A = final amount
P = initial amount
r = rate
n = years or time
Therefore,
A = 12000(1-0.028)¹²
A = 12000(0.972)¹²
A = 8534.46
Hence, the depreciated value of the bike after 12 years is $8534.46
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Use distributive property to rewrite each algebraic expression: 4(10 + x
Answer:40+4x
Step-by-step explanation:
1) distribute the 4 to each term
4(10+x)
40+4x
Jocelynn rode her motorcycle at an average speed of 45 miles per hour for 4.5 hours then rode at an average speed of 50 per hour for 2.5 hours. What was the total distance in miles that Jocelynn rode her motorcycle during this time?
Answer:
327 miles
Step-by-step explanation:
Answer:
327.5 miles
Step-by-step explanation:
45 times 4.5 + 50 times 2.5 = 327.5
Help pleaseee Double points
Answer: y=17.3
refer to the image for the steps
If m∠5 = x°, which angles also have a measure of x°? Select all that apply.
The angle that are equal to m∠5 are ∠9 and ∠12, therefore, C and D.
What is an angle?When two straight lines or rays intersect at a shared endpoint, an angle is generated. An angle's vertex is the common point of contact. Angle is derived from the Latin word angulus, which means "corner."
As per the given image, the two streets 1st ave and 2nd ave are parallel to each other while D street acts as a transverse and intersects both 1st ave and 2nd ave. Therefore, the angle that will be equal to m∠5 are:
m∠5 = m∠9, Becuase the two angles are the corresponding angles
m∠5 = m∠9m = m∠12, Becuase ∠9 and ∠12 are vertically opposite angles.
Hence, the correct options are C and D.
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What is one over three to the fifth power?
six over eight
five over fifteen
one over one hundred twenty-five
one over two hundred forty-three
What are the domain and range of g of x equals the square root of the quantity x minus 3?
D: [3, ∞) and R: [0, ∞)
D: [–3, ∞) and R: [0, ∞)
D: (–3, ∞) and R: (–∞, 0)
D: (3, ∞) and R: (–∞, 0)
Answer:
90-99=86x45[80[
Step-by-step explanation:
Answer:
The function g(x) = √(x - 3) is defined for all real numbers x such that x - 3 ≥ 0, since the square root of a non-negative number is a real number.
Solving for x, we have:
x - 3 ≥ 0
x ≥ 3
Therefore, the domain of the function is D: [3, ∞).
Since the square root of a non-negative number is always non-negative, the range of the function is R: [0, ∞).
Therefore, the correct answer is: A. D: [3, ∞) and R: [0, ∞).
How do you find the x and y coordinates of the vertex of a parabola?.
The process of find the x and y coordinates of the vertex of a parabola is y = -b/2a
Coordinates in math is known as a number on the x-axis called an x-coordinate and a number on the y-axis called a y-coordinate.
Here we need to write down the process of finding the x and y coordinates of the vertex of a parabola.
The first step is to Get the equation in the form y = ax² + bx + c.
And then we have to calculate -b / 2a ad this is the x-coordinate of the vertex.
In order to find the y-coordinate of the vertex, then we have to simply plug the value of -b / 2a into the equation for x and solve for y.
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HELP ASAP PLSSSSS
Part A: Given the function g(x) = |x − 7|, describe the graph of the function, including the vertex, domain, and range. (5 points)
Part B: If the parent function f(x) = |x| is transformed to h(x) = |x| + 2, what transformation occurs from f(x) to h(x)? How are the vertex and range of h(x) affected?
Part A:
The graph of the function g(x) = |x - 7| is a V-shaped graph that opens upwards and has a vertex at x = 7. The vertex is the midpoint of the graph and occurs at the value of x where the absolute value changes from positive to negative. The domain of the function is all real numbers and the range is all non-negative numbers.
Part B:
The transformation from f(x) to h(x) occurs by shifting the graph of the parent function up by 2 units along the y-axis. In other words, every y-coordinate in the graph of f(x) is increased by 2 in the graph of h(x). The vertex of the parent function is (0,0) and is shifted to (0,2) in the transformed function. The range of h(x) is all non-negative numbers greater than or equal to 2.
A garden table and a bench cost $621 combined. The garden table costs $79 less than the bench. What is the cost of the bench?
Answer:
$389.5
Step-by-step explanation:
621/2 = 310.5 assuming they're equal therefore, it is stated that the table cost cost 79 less than the bench so we substract.
310.5 - 79 = 231.5
This is the cost of garden table, lets find the bench now.
621(both table and bench)-231.5(garden table)= 389.5 is the cost of the bench.
Uing the following table, what term bet decribe the entry of 62%?
Which location would you mot like to viit?
Apen, Colorado New York,
New York Row total
Male tudent 22% 28% 50%
Female tudent 16% 34% 50%
Column total 38% 62% 100%
Group of anwer choice
Joint frequency
Joint relative frequency
Marginal frequency
Marginal relative frequency
Answer:
50%
Step-b50%y-step explanation:
Joint relative frequency
When given three side measures of a triangle, when will you be able to create a unique triangle, no triangle, and/or many triangles? Explain you reasoning.
The key is to check whether the given side measures satisfy the triangle inequality theorem or not, if yes then we can create a unique triangle, if no then no triangle can be created, and if the given measures are the perimeter of a triangle, we can create many triangles.
When will you be able to create a unique triangle, no triangle, and/or many triangles?When given three side measures of a triangle, the following scenarios can occur:
Unique triangle: If the three side measures given are the lengths of the three sides of a triangle and they satisfy the triangle inequality theorem, then we can create a unique triangle. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
No triangle: If the three side measures given do not satisfy the triangle inequality theorem, then it is not possible to create a triangle with those side lengths.
Many triangles: If the three side measures given are the perimeter of a triangle, and we are allowed to vary the lengths of the three sides of the triangle within a certain range, then we can create infinitely many triangles.
Example: A triangle with perimeter 12cm, we can create a unique triangle with sides 3,4,5, or we can create many triangle with sides (2,5,5), (4,4,4),(3,4,5) and so on.
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Rewrite the entence, adding a nonretrictive element uing parenthee hi book mut be intereting becaue he han’t put it down once
The revised form of the sentence would be as follows: Mark Twain, who was born in Florida, is one of America's most famous authors.
Nonrestrictive elements are sets of words that can be deleted from a statement without affecting its meaning.
A nonrestrictive element is nonessential because we don't need the information it provides. These elements are always separated with commas from the rest of the sentence to specify that they are not essential. If they were dropped, the meaning of the sentence would be left unchanged.
Mark Twain, is one of America's most famous authors. - In the above sentence the nonrestrictive element is a nonessential relative clause.
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